I am trying to define a continuous branch of log z in the

complex plane slit along the positive axis:C\{[0,i*infinity]}

I understand that we need to make a branch cut because log z will not be continuous on all ofC. When the cut is the negative real axis I think I understand. And I also realize that the the cut can be any ray by my intuition. But I don't understand how to put it down as a function in this case.